<?xml version="1.0"?><ownershipDocument><schemaVersion>X0201</schemaVersion>    <documentType>4</documentType><periodOfReport>2002-02-27</periodOfReport><notSubjectToSection16>0</notSubjectToSection16><issuer><issuerCik>0000914024</issuerCik><issuerName>SHAW GROUP INC</issuerName>        <issuerTradingSymbol>SGR</issuerTradingSymbol></issuer><reportingOwner><reportingOwnerId><rptOwnerCik>0001064343</rptOwnerCik><rptOwnerName>HOYLE DAVID H</rptOwnerName></reportingOwnerId><reportingOwnerAddress><rptOwnerStreet1>316 S. MARIETTA STREET</rptOwnerStreet1><rptOwnerStreet2></rptOwnerStreet2><rptOwnerCity>GASTONIA</rptOwnerCity><rptOwnerState>NC</rptOwnerState><rptOwnerZipCode>28052</rptOwnerZipCode><rptOwnerStateDescription></rptOwnerStateDescription></reportingOwnerAddress><reportingOwnerRelationship><isDirector>1</isDirector><isOfficer>0</isOfficer><isTenPercentOwner>0</isTenPercentOwner><isOther>0</isOther></reportingOwnerRelationship></reportingOwner><nonDerivativeTable><nonDerivativeTransaction><securityTitle><value>COMMON STOCK</value></securityTitle><transactionDate><value>2002-02-27</value></transactionDate><deemedExecutionDate></deemedExecutionDate><transactionCoding><transactionFormType>4</transactionFormType><transactionCode>G</transactionCode><equitySwapInvolved>0</equitySwapInvolved></transactionCoding><transactionTimeliness><value></value></transactionTimeliness><transactionAmounts><transactionShares><value>250</value></transactionShares><transactionPricePerShare><footnoteId id="F1" /></transactionPricePerShare><transactionAcquiredDisposedCode><value>A</value></transactionAcquiredDisposedCode></transactionAmounts><postTransactionAmounts><sharesOwnedFollowingTransaction><value>33250</value><footnoteId id="F2" /></sharesOwnedFollowingTransaction></postTransactionAmounts><ownershipNature><directOrIndirectOwnership><value>I</value><footnoteId id="F2" /></directOrIndirectOwnership><natureOfOwnership><value>By spouse</value><footnoteId id="F2" /></natureOfOwnership></ownershipNature></nonDerivativeTransaction></nonDerivativeTable><footnotes><footnote id="F1">No price, because it was a gift by inheritance from mother's estate. </footnote><footnote id="F2">33,250 shares total, 27,000 shares of which are directly owned and 6,250 are indirectly owned by spouse.</footnote></footnotes><remarks></remarks><ownerSignature><signatureName>/s/ David W. Hoyle</signatureName><signatureDate>2003-10-30</signatureDate></ownerSignature></ownershipDocument>

